Presentation on theme: "Online Social Networks and Media Navigation in a small world."— Presentation transcript:
Online Social Networks and Media Navigation in a small world
Small world phenomena Small worlds: networks with short paths Obedience to authority (1963) Small world experiment (1967) Stanley Milgram (1933-1984): “The man who shocked the world”
Small world experiment Letters were handed out to people in Nebraska to be sent to a target in Boston People were instructed to pass on the letters to someone they knew on first-name basis The letters that reached the destination followed paths of length around 6 Six degrees of separation: (play of John Guare)
Milgram’s experiment revisited What did Milgram’s experiment show? – (a) There are short paths in large networks that connect individuals – (b) People are able to find these short paths using a simple, greedy, decentralized algorithm
Small worlds We can construct graphs with short paths – E.g., the Watts-Strogatz model
Navigation in a small world Kleinberg: Many random graphs contain short paths, but how can we find them in a decentralized way? In Milgram’s experiment every recipient acted without knowledge of the global structure of the social graph, using only – information about geography – their own social connections
Kleinberg’s navigation model Assume a graph similar (but not the same!) to that of Watts-Strogatz – There is some underlying “geography”: ring, grid, hierarchy Defines the local contacts of a node Enables to navigate towards a node – There are also shortcuts added between nodes The long-range contacts of a node Similar to WS model – creates short paths
Kleinberg’s navigational model Given a source node s, and a navigation target t we want to reach, we assume – No centralized coordination Each node makes decisions on their own – Each node knows the “geography” of the graph They can always move closer to the target node – Nodes make decisions based only on their own contacts (local and long-range) They do not have access to other nodes’ contacts – Greedy (myopic) decisions Always move to the node that is closest to the target.
Theorem: For α =1 there is a polylogarithimic search algorithm. For α ≠1 there is no decentralized algorithm with poly-log time – note that α =1 and the exponential dependency results in uniform probability of linking to the subtrees
Application: P2P search -- Symphony Map the nodes and keys to the ring Link every node with its successor and predecessor Add k random links with probability proportional to 1/(dlogn), where d is the distance on the ring Lookup time O(log 2 n) If k = logn lookup time O(logn) Easy to insert and remove nodes (perform periodical refreshes for the links)
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